Stone-Weierstrass theorem for homogeneous polynomials and its role in convex geometry

TL;DR

This paper proves a conjecture related to uniform approximation of boundary characteristic functions of convex sets using homogeneous polynomials, and introduces a new measure called the d-volume ratio with bounds, advancing convex geometry theory.

Contribution

It establishes the Stone-Weierstrass theorem for homogeneous polynomials and introduces the d-volume ratio with new bounds, connecting polynomial approximation with convex geometry.

Findings

Proves the conjecture by Kroo on polynomial approximation of convex set boundaries.

Introduces the d-volume ratio and provides upper bounds for it.

Demonstrates the approximation rate with explicit error bounds.

Abstract

We give a uniform approximation of the characteristic function of the boundary of a centrally symmetric n-dimensional compact and convex set by homogeneous polynomials of even degree $d$ fulfilling $|g_d-1|\leq E/d^{1/2-\beta}$, for every $\beta>0$, large enough $d$, and some constant $E$ only depending on $n$ and $K$. In particular, this proves a conjecture posed by Kroo in 2004, also known as the Stone-Weierstrass theorem for homogeneous polynomials. Moreover, we introduce the d-volume ratio for a convex body $K$ in $\mathbb R^n$, by means of its d-Lasserre-L\"owner polynomial. We also prove an upper bound of the d-volume ratio of the form $1+F/d^{3/2-\beta}$, for every $\beta>0$, large enough $d$, and $F$ some constant only depending on $n$.